Question 1208773
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Premise:
1. (L ≡ N) ⊃ C
2. (L ≡ N) ∨ (P ⊃ ~E)
3. ~E ⊃ C
4. ~C
Conclusion: ~P
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Here is one way to do the derivation. There may be other approaches.
<table border = "1" cellpadding = "5"><tr><td>Number</td><td>Statement</td><td>Line(s) Used</td><td>Reason</td></tr><tr><td>1</td><td>(L ≡ N) ⊃ C</td><td></td><td></td></tr><tr><td>2</td><td>(L ≡ N) v (P ⊃ ~E)</td><td></td><td></td></tr><tr><td>3</td><td>~E ⊃ C</td><td></td><td></td></tr><tr><td>4</td><td>~C</td><td></td><td></td></tr><tr><td>:.</td><td>~P</td><td></td><td></td></tr><tr><td>5</td><td>~(L ≡ N)</td><td>1,4</td><td>Modus Tollens</td></tr><tr><td>6</td><td>P ⊃ ~E</td><td>2,5</td><td>Disjunctive Syllogism</td></tr><tr><td>7</td><td>~(~E)</td><td>3,4</td><td>Modus Tollens</td></tr><tr><td>8</td><td>E</td><td>7</td><td>Double Negation</td></tr><tr><td>9</td><td>~P</td><td>6,8</td><td>Modus Tollens</td></tr></table>
Here's a list of <a href="https://www.algebra.com/algebra/homework/Conjunction/logic-rules-of-inference-and-replacement.lesson">rules of inference and replacement</a>
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